All of the 5th grade teachers and students from Oak went on a field trip to an art museum. Tickets were $$5.00$ each for teachers and $$2.50$ each for students, and the group paid $$45.00$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$20.00$ each for teachers and $$9.50$ each for students, and the group paid $$175.00$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5x+2.5y = 45}$ ${20x+9.5y = 175}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-20x-10y = -180}$ ${20x+9.5y = 175}$ Add the top and bottom equations together. $ -0.5y = -5 $ $ y = \dfrac{-5}{-0.5}$ ${y = 10}$ Now that you know ${y = 10}$ , plug it back into $ {5x+2.5y = 45}$ to find $x$ ${5x + 2.5}{(10)}{= 45}$ $5x+25 = 45$ $5x = 20$ $x = \dfrac{20}{5}$ ${x = 4}$ You can also plug ${y = 10}$ into $ {20x+9.5y = 175}$ and get the same answer for $x$ ${20x + 9.5}{(10)}{= 175}$ ${x = 4}$ There were $4$ teachers and $10$ students on the field trips.